Homework Help: Optimization Problem - Calculus

1. Mar 4, 2010

polak333

1. The problem statement, all variables and given/known data

A cylindrical shaped tin can must have a volume of 1000cm3.

Find the dimensions that require the minimum amount of tin for the can (Assume no waste material). The smallest can has a diameter of 6cm and a height of 4 cm.

2. Relevant equations

$$V = \pi r^{2}h$$

$$P = 2( \pi r^{2}) + (2 \pi r)h$$

3. The attempt at a solution

I basically start off by isolating a variable:

$$V = \pi r^{2}h$$

$$1000 = \pi r^{2}h$$

$$318.3 = r^{2}h$$

$$r^{2} = \frac{318.3}{h}$$
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This is where I'm stuck. Do I plug this into the surface area formula.
I know I have to sub this into something else, and then expand and find the derivative of that new equation to find where h > 4.
Just need to find out what to do next from here.

Thanks for any help!

2. Mar 4, 2010

Staff: Mentor

You have a relationship between r and h (but leave pi in - 1000/pi is not exactly equal to 318.8). Now rewrite your surface area function P (why is it P?) so that it is a function of one variable, either r or h. Then use the normal technique for finding the minimum or maximum function value.